Abstract: Methods and apparatus to present solutions to problems are disclosed. An example method includes receiving a set of rules defining a problem, electronically propagating a first solution to determine a second solution and a third solution, automatically determining if a cost associated with the third solution is less than a cost associated with the second solution, storing the cost associated with the third solution as a first minimum value using an atomic operation when the cost associated with the third solution is less than the cost associated with the second solution, and displaying the third solution.

Abstract: A method for storing bigram word indexes of a language model for a consecutive speech recognition system (200) is described. The bigram word indexes (321) are stored as a common two-byte base with a specific one-byte offset to significantly reduce storage requirements of the language model data file. In one embodiment the storage space required for storing the bigram word indexes (321) sequentially is compared to the storage space required to store the bigram word indexes as a common base with specific offset. The bigram word indexes (321) are then stored so as to minimize the size of the language model data file.